Rational H2 approximation
Since finite order LTI systems and their rational transfer functions are the most used models in system theory, rational approximation is at the heart of modeling problems. The challenge is to select a model that is close enough to a physical system and yet simple enough to be studied analytically. Modeling is thus at the origin of a fertile interaction between rational approximation methods and system theory. A great number of methods have been proposed in the literature, that divide into two main groups: projection methods and optimization methods.
Among these methods, rational approximation in the Hardy space H2 presents a number of interesting features. Assuming a transfer function belongs to H2 corresponds to a stability condition for the underlying physical system: a bounded energy input produces a bounded output. The Hardy space H2 possesses a very rich structure which combines analyticity properties and an Hilbert space framework. However, H2 rational approximation is a difficult non-linear problem due to the complexity of the set approximants (rational matrices of fixed order) and the existence of many local minima.
We have developped a specific approach to cope with this problem, which is based on the following points
The softare RARL2 has been developped following this approach.
- A compactification of the optimization set which makes use of the Douglas-Shapiro-Shields factorization and the projection theorem in Hilbert space. Lossless matrix valued functions play in the matrix case the role of the denominator in the scalar case. The "denominator" is thus optimized while the "numerator" is computed by projection: the optimization runs over the set of lossless functions which thus enter the picture and bring their rich structure.
- The use of an atlas of charts to describe the optimization set: the space of lossless functions of fixed McMillan degree. This type of representation is recommended if one wants to use differential calculus tools for solving an optimization problem. Schur interpolation theory provides such atlases of charts.